Fig. 1

An illustration of the CDFs discussed. The graph illustrates the distributions of the variables \( W \) and \( X \) using a standard log-normal CDF to depict the positive part of \( H(x) \), represented by the blue line, with \( {\pi}^{*} \) set to 0.1. The assumptions on \( H(x) \) imply that \( G(x) \), represented by the green line, is continuous and increasing over \( x>\delta \) with \( \delta \) set to 0.5. For \( x\le \delta \), \( F(x)=\pi \) which is equal to 0.32 in this illustration, represented by the pink line. For \( x>\delta \), \( F(x)= H(x) \), but their graphs were slightly shifted for clarity